Best Known (81, 81+26, s)-Nets in Base 4
(81, 81+26, 1028)-Net over F4 — Constructive and digital
Digital (81, 107, 1028)-net over F4, using
- 1 times m-reduction [i] based on digital (81, 108, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 27, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 27, 257)-net over F256, using
(81, 81+26, 1294)-Net over F4 — Digital
Digital (81, 107, 1294)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4107, 1294, F4, 26) (dual of [1294, 1187, 27]-code), using
- 254 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 21 times 0, 1, 34 times 0, 1, 48 times 0, 1, 57 times 0, 1, 65 times 0) [i] based on linear OA(496, 1029, F4, 26) (dual of [1029, 933, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(496, 1024, F4, 26) (dual of [1024, 928, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(491, 1024, F4, 25) (dual of [1024, 933, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- 254 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 21 times 0, 1, 34 times 0, 1, 48 times 0, 1, 57 times 0, 1, 65 times 0) [i] based on linear OA(496, 1029, F4, 26) (dual of [1029, 933, 27]-code), using
(81, 81+26, 170480)-Net in Base 4 — Upper bound on s
There is no (81, 107, 170481)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 26328 491718 872231 780766 803360 018017 024140 212441 253198 592293 701440 > 4107 [i]